In the figure both Δ ABC and Δ DCX are equilateral, 'D' is the midpoint of BC and AX intersects BC at Y. Show that BY = 2YC. Share with your friends Share 2 Manbar Singh answered this Since ∆ABC is an equilateral triangle, then∠ABC = ∠BCA = ∠ACB = 60°Also , AB = BC = CASince, ∆DCX is an equilateral triangle, then∠DCX = ∠DXC = ∠XDC= 60°Also , DC= CX = XDIn ∆AYB and ∆XYC∠ABY = ∠XCY 60° each∠AYB = ∠XYC Vertically opposite angles⇒∆AYB and ∆XYC AA⇒AYXY = BYCY = ABXC Corresponding sides of similar ∆'s are proportional⇒BYCY = ABXC ....1Let AB = BC = CA = 2xNow, D is the mid point of BC, then BD = DC = BC2 = xSince, DC = XC = XD, soDC = XC = XD = x as, DC = xNow, ABXC = 2xx = 21So, from 1, we getBYCY = ABXC = 21⇒BYCY = 21⇒BY = 2CY 2 View Full Answer Aby Joseph Biju answered this figure in the link below 0 Aby Joseph Biju answered this sorry link not getting uploaded photo (figure) is here 3 The Mykey Siddharth answered this SO in triangle ABC, Angle of A = angle of B = angle of C = 60 degree. Take AB = AC = BC = 2x --------- (each side of equilateral triangle is same).................(1) In triangle DCX, Angle of D = angle of C = angle of X = 60 degree. Since D is the mid point of BC, this implies DC = CX = DX = 2x / 2 = x .......................(2) Now triangle AYB is similar to triangle XYC (AAA similarity, angle CYX is equal to angle AYB, angle YCX is equal to YBA and angle YXC is equal to YAB) Therefore, AY / XY = YB / YC = AB / XC ------ (property of similar triangle) Now, => BY / YC = AB / XC => BY / YC = 2x / x --------- ( from 1 and 2 ) => BY / YC = 2 => BY = 2 YC. Hope it helps ...!! 6
